Czechoslovak Mathematical Journal Daniel Gluschankof Cyclic ordered groups and MV-algebras Czechoslovak Mathematical Journal, Vol. 43 (1993), No. 2, 249–263 Persistent URL: http://dml.cz/dmlcz/128391 Terms of use: © Institute of Mathematics AS CR, 1993 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz Czechoslovak Mathematical Journal, 43 (118) 1993, Praha CYCLIC ORDERED GROUPS AND MV-ALGEBRAS DANIEL GLUSCHANKOF, Angers and Paris (Received September 10, 1991) In the forties and fifties two—at the moment—unrelated concepts derived from that of an ordered group appeared. The notion of cyclic-ordered group (c-group) (see [9], [10], [13] and [14]) and that of MV-algebra (see [4] and [5]). The first one appeared as a way of generalizing the notion of totally ordered groups. That notion was further extended to that of partially cyclically ordered groups. The notion of MV-algebras resulted from a succesfull attempt of giving an algebraic structure to the infinite-valued Lukasievicz propositional logics. In the last decade, that theory was fruitfully linked with that of a class of C*-algebras (see [8]). The objective of this work is to show that suitable subclasses of that notions can be linked by the way of a covariant functor. 1. DEFINITIONS AND FIRST FACTS A cyclically ordered group (c-group) is a system (G, +, —, 0,T) where (G, +, —, 0) is a group (not necessarilly commutative) and T is a ternary relation verifying the following properties: CI. Va6c (if a -^ 6 -^ c 7- a then exactly one of T(a^ 6, c) and T(a, c, 6) holds); C2. Va6c (T(a, 6, c) => a £ b ± c ± a); C3. Va6c (T(a, 6, c) => T(c, a, 6)); C4. Va6cd (T(6, c, a) k T(c, d, a) => T(6, d, a)); C5. Va6cd (T(a, 6, c) =t> T(d + a, d + 6, d + c) k T(a + a1, 6 + a1, c + d)). A fundamental result of Rieger (see [9]) says that any such a group is isomorphic to a quotient of a totally ordered group (o-group) by the subgroup generated by a strong unit (a cofinal element in its centre). In that case, if G = (G, +, —, 0, w, -$) is an o-group with strong unit u, the quotient group Gu = G/(u) can be endowed with a cyclic order by defining T(a,6,c) if and only if, for the only representatives a, 6, c such that 0 $C a, 6, c < uy either a<6<cor6<c<aorc<a<6 holds. 249 The notion of c-group generalizes that of totally ordered groups (o-groups) in the sense that for a c-group with the property: for all a £ G} T(—a,0,a) implies, for all n € N, T(—na,0, na) a total order (compatible with the group operation) can be defined by 0 < a if and only if T(—a,0,a). Conversely, an o-group can be endowed with a c-group structure by defining T(a, 6, c) if and only ifa<6<cor6<c<a or c < a < 6. A partially cyclically ordered group (pco-group) is a system (C,+, —,0,T) where the axioms C3, C4, C5 and Clp. Va6c(T(a,6,c)=>-1T(a,c,6)); C6. Va6c(T(a, 6, c) => T(-c, -6, -a)) hold. This last axiom is consequence of axioms CI . • .C5 and C2 is consequence of Clp and C3. Observe that, Rieger's theorem also holds in this case by replacing the o-group by a partially ordered group (po-grup) (see [13] or [14]). An M V-algebra (see [4], [5] and [8]) is a system (-4, ©,*,-», 0,1) which satisfies the following universal identities: mi xф(yфz) = (xфy)фz ITІ2 xфQ = x m3 xфy=yфx m.4 xф 1 = 1 -i-iX =. x m5 m6 -Ю = 1 ПI7 x ф -.x = 1 m8 -•(-•* Фy)Фy = -i(xф^y)фx m9 x*y = -Ҷ-чe ^y) By defining iVj/:=(.c* -»t/) ££ y and, by duality, x A y := -i(-»x V -»t/) we have that (.A, V, A,0,1) is a bounded distributive lattice. Another approach for this structures is that of Wajsberg algebras (W-algebras) (see [6] and [11]). Such an algebra is a system {A, —•, -i, 0,1) satisfying the following 250 universal identities: Wl. (x - y) - ((y -> z) - (x -* *)) = 1; W2. (a: — y) —• y = (y -+ x) —• x; W3. (-»x —> -iy) —* (y —• x) = 1; W4. l-->x = x; W5. x -* 0 = -•*; W6. -M = 0; W7. -.0=1. By defining x V y := (x —• y) —• y and x A y := -»(-»x V -«y) (.4, V, A, 0,1) results also a bounded distributive lattice. In [6] it is proved that a W-algebra can be thought of as an MV-algebra (and viceversa) by identifying the respective 0,1 and -» and defining: a —• b := -»a 0 6 and a 0 b := -ia —• 6; (recall that the operation * of the MV-algebra can be defined in terms of 0 and -»). In [4] it is proved that any MV-algebra A can be obtained from an abelian lattice- ordered group (1-group) with strong unit u G = (G, V, A,+,— ,0, u) by defining: A = [0, u] = {a | 0 ^ a ^ u); a 0 6 = (a + 6) A w; -ia = u — a and 1 = «. Since any MV-algebra derives from an abelian 1-group, in the sequel group will stand for abelian group, homomorphism and subgroup for homomorphism and sub­ group for the respective structures (o-groups, c-groups, pco-groups, 1-groups, MV- algebras). 2. LATTICE PCO-GROUPS For any pco-group G, a partial order be defined by (*) a ^ b if and only if a = 6 or T(0, a, b) or a = 0. This order makes every element "positive". Observe that, in general, .$ is not compatible with the group operation, for example, by setting G = Z/3Z with its natural cyclical order, the total order (*) induced is given by the set of pairs {(0,0), (1,1), (2,2), (0,1), (0,2), (1,2)} which is obviously non-compatible, since 1 ^ 2 holds but 2=1 + 1^2+1 = 0 does not hold. 251 We say that a group homomorphism /: G —• H between pco-groups is a pco- homomorphism if, for a,6,c G G such that T(a,6,c), if/(a) ^ f(b) -/ f(c) ^ f(a) then T(/(a),/(6),/(c)). Observe that a pco-homomorphism is also a homomorphism for the order given in (*). Definition 2.1. A pco-group G will be called a lattice-cyclical-group (and de­ noted lc-group), if, for the order defined in (*) the structure (G,0,i$) admits a distributive lattice structure with first element. Lemma 2.2. Let G be an lc-group, a, 6 G G. If a ^. a + b (b ^. a + b) then 6 ^ a-f 6 (a -$ a -f b), implying a V 6 :$ a -f 6. Proof. Suppose 0 < a < a -f 6 (the other cases are immediate). Then we have T(0, a, a -f 6), which, adding —(a -f 6) to each term, implies T(—(a -f 6), —6,0) which, by axiom C6, is equivalent to T(0, 6, a -f 6), proving our claim. D Definition 2.3, Let G be an lc-group and H a subgroup. (i) II is called an lc-ideal if it is convex for the order .$ (that is, for all x G H, z G G, z ^ x implies z G H), and is an 1-subgroup (that is, for x, y G H, xWy G H). (ii) II is called a pc-subgroup if it is convex for the relation T (that is, for x, y G H and z G G, T(x, z, y) implies z G G). Observe that the lc-ideals (pc-subgroups) are the kernels of lc(pc)-homomor- phisms. Moreover, the lc-ideals are lattice-ideals for the structure (G,0, V, A). Ob­ serve also that for cyclically ordered groups, the T-convex subgroups are always trivial. Lemma 2.4. Let G be an lc-group and H a subgroup. H is T-convex if and only if it is ^-convex. So, any pc-subgroup preserving the lattice operations is also an lc-ideal. Proof. Let H be T-convex, a e H, b e G such that 0^6^ a. If 6 = 0 or 6 = a, it is immediate that 6 G H. So we can write T(0,6,a), implying, by T-convexity, that 6 G H. For the converse, if H is ^-convex, a,c G H, 6 G G such that T(a, 6, c). By axiom C5 we have T(0, 6 — a, c — a). Since II is ^-convex, we conclude that 6 — a G Hand then6GH. • So, without abuse of notation, we can speak about convex subgroups. Lemma 2.5. Let G be an lc-group, H C G a.* lc-ideal. H is prime if and only if the quotient G/H is cyclically ordered. 252 Proof. By a result on distributive lattices (see [1, II1.3]) we have that the lattice (G///,0,V,A) ~ (G,0,V,A)/H is totally ordered if and only if H is prime as a lattice ideal.